# Monomials in products in power series ring on several variables

Let $$A \colon= K[[X_1,\ldots,X_m,Y_1,\ldots,Y_n]]$$ be a power series ring over a field $$K$$ in $$m + n$$ variables and $${\frak m}$$ be the unique maximal ideal of $$A$$.

For arbitrary two elements $$\alpha = X_1^{e_1} \cdots X_m^{e_m}$$, $$\beta = Y_1^{f_1} \cdots Y_n^{f_n}$$ with $$e_i, f_j \geq 1$$, we shall add $$a \in {\frak m}$$ to $$\alpha$$ and $$b \in {\frak m}$$ to $$\beta$$ such that $$a$$ (resp. $$b$$) comprises scalar times monomials different from $$\alpha$$ (resp. $$\beta$$). Consequently, $$a + \alpha \in {\frak m}$$ (resp. $$b + \beta \in {\frak m}$$) contains $$\alpha$$ (resp. $$\beta$$) as its non-trivial constituent.

Q. Does the following sufficiently large constant $$C \gg 0$$ which is independent of $$m, n$$ exist? :

The product $$(a + \alpha)(b + \beta)$$ always involves a nonzero monomial $$X_1^{v_1} \cdots X_m^{v_m}Y_1^{w_1} \cdots Y_n^{w_n}$$ (up to scalar coefficient) such that inequalities $$v_i \leq Ce_i$$, $$w_j \leq Cf_j$$ for $$1 \leq i \leq m$$ and $$1 \leq j \leq n$$ hold simultaneously.

Yes if you allow $$C$$ to depend on $$n$$ and $$m$$. Define the graded degree of a monomial$$\prod X_i^{v_i}\prod Y_j^{w_j}$$ as $$\sum v_i/e_i+\sum w_j/f_j$$. Then the first bracket $$a+\alpha$$ contains the monomial $$\alpha$$ with graded degree at most $$m$$, the second bracket $$b+\beta$$ contains the monomial $$\beta$$ with graded degree at most $$n$$, thus the product $$(a+\alpha)(b+\beta)$$ contains a monomial with graded degree at most $$m+n$$ (the products of monomials of minimal graded degree do not cancel.) This proves your claim for $$C=m+n$$.
• Great thanks! But I wish C to be independent of $m, n$. Rinmyaku – Rinmyaku Jul 13 at 10:10